optionpricing

Option pricing refers to the process of determining the fair value of options, which are financial instruments that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price before or at expiration. Prices depend on the current asset price, strike price, time to expiration, volatility, interest rates, and dividends.

There are call options (right to buy) and put options (right to sell). The intrinsic value is max(S-K,0) for a call, max(K-S,0) for a put; time value reflects potential for favorable moves before expiration. Put-call parity links prices of calls and puts on the same asset.

The Black-Scholes model provides a closed-form solution for European options on non-dividend-paying assets, giving a formula for calls and put prices. The Cox-Ross-Rubinstein binomial model provides a discrete-time, recombining lattice that can handle dividends and American options. Most practical pricing uses a combination of models, numerical methods, and market data.

Numerical methods include Monte Carlo simulation and finite difference methods solving the Black-Scholes partial differential equation. Market practice also uses implied volatility, creating a volatility surface or smile that reflects varying volatilities across strikes and maturities.

Assumptions include no arbitrage, frictionless markets, lognormal asset returns, constant volatility and interest rates. Model risk arises when these assumptions fail. American options may require early-exercise considerations, making pricing more complex.

Prices are used for trading, hedging, and risk management, with Greeks providing sensitivity measures to changes in inputs. Dividends can be incorporated via adjusted asset price or yield, and the field continues to evolve with new models and numerical techniques.